# How does binding energy work?

I do not have a very advanced understanding of binding energy and atomic mass but in my classes, I have learned that the atomic mass is equal to the sum of the number of protons and the number of neutrons. But I know this to not be the case every time because during fusion energy is emitted due to the mass of the product (say, Helium-4) being lighter than the nuclei that you started with (say, 2 protium nuclei). In a book (The Universe in Your Hand, by Christophe Galfard), I read that the difference in mass from which the energy is emitted (due to E=mc^2) is actually from the mass of the mesons (two quarks?) and gluons that carry the strong force between the quarks and between protons and neutrons. According to the book, when fusion occurs the number of gluons and mesons required to hold the atom together is less (the binding energy?), and therefore the total mass decreases.

Now my question is, why does this fusion cause a decrease in mass? For example, why does a hydrogen-2 nucleus have less binding energy, i.e. have fewer mesons and gluons, than two hydrogen-1 nuclei? The hydrogen-1 nuclei should not require any mesons since they only consist of a proton, don't they? If someone could explain this I would be very grateful.

Please inform me of potential errors. :)

Binding Energy is considered negative because it's the energy needed to separate the constituent parts.

Suppose you have a simple Hydrogen atom, just a proton and an electron. You need 13.6eV to separate the proton and electron an infinite distance away from each other.

So the energy of the atom in the ground state is $$m_pc^2+m_ec^2+BE$$ where BE=-13.6 eV. Note the minus sign.

So the energy does roughly increase with the rest mass of the constituent parts because the binding energy is orders of magnitude smaller than the constituent parts.

Now suppose the hydrogen atom has a neutron in addition to the proton. Then we get new terms in the sum of the energy. We have to add $$m_nc^2$$ for the mass energy of the neutron. But then the neutron is bonded to the proton via the strong force, we have a new Binding Energy term. There's also some angular momentum terms now that might not show up in the individual rest masses or the binding energy.

So we expect a smaller mass-energy for particles bound together because we need to add energy to break them up into constituent parts. The need to add to get to the total rest energy of the parts suggests a deficit, so the binding energy is negative.

When the nucleons are grouped together to form a nucleus, they lose a small amount of mass, i.e., there is a mass defect. This mass is typically associated with the binding energy between nucleons.

The "missing" mass is the energy released by the formation of the atomic nucleus.

Now, for why this exists, you know that you need to put some energy or expend some energy to split the atom apart into its components, else all the atoms around us would just spontaneously break up (if they were put together-or rather, they would never even form in the first place.) This is similar to when two objects are gravitationally bound: you need to expend some energy to separate them.

The difference in mass between the reactants and products is manifested as either the release (when hydrogen/helium are fused) or the absorption of energy(when heavier elements are fused, it requires energy from the source).

This difference in mass arises due to the difference in atomic binding energy between the nuclei before and after the reaction. Suppose you have two hydrogen atoms apart. Now you compare it to a helium atom. Obviously, to bring these two hydrogens together, one needs to take energy from them (as derived from theory {which I won't be able to explain here} as well as confirmed by observation of the existence of atoms which depends entirely on there existing a Binding Energy, THUS THEY NOT SPONTANEOUSLY SPLITTING) and to tear the helium apart, we need to make it undergo fission, which is endothermic (we need to give it energy), so the hydrogen must have lost some energy.

So, that means by E=mc^2 that the nucleons of hydrogen atoms have lost mass to come together which implies that the rest mass of two separate hydrogen atoms is more than the helium atom. SO, WHILE BEING TIGHTLY BOUND, THE NUCLEUS LOSES ENERGY CAUSING IT TO LOSE MASS RELATIVE TO ITS CONSTITUENT NUCLEI (OBTAINED AFTER FISSION).